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A General Overview of Fluid Mechanics 
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Fluid Mechanics 101
"A General Overview of Fluid Mechanics and How It Pertains To Everything In Your Life"

Foreword: I apologize for the delay in writing this. As soon as the voting results came in, my life became extremely hectic for a solid month or two. It's been a crazy time, but things are calming down. Now, as requested, my overview of fluid mechanics. PLEASE, for the love of all that is good and holy, ask questions! Challenge any assumptions I may have made! If you're confused about something, ask! I would rather be wrong and learn something new than to be wrong and not realize it at all; or worse yet, be right and become content in my understand.




Introduction

I would imagine the first thing to pop into your head is "how does fluid mechanics pertain to everything in my life?" Well okay, maybe I exaggerated a little bit. We can, of course, survive in the world without having to understand the fluid processes that take place when we boil a pot of water, drive down the road, or take a ride in an airplane to go see your Great Aunt Flo (see what I did there? If you didn't, you will soon enough). However, understanding basic fluid processes can open your eyes and let you see the world in a way that you never imagined.

So let's get down to business. Starting off: what is a fluid?

From the engineer's perspective, there are really only two types of matter in the universe: solids and fluids. Liquid, gas, or even plasma can all fall under the definition of a "fluid". The two are easily discernible to the layperson, but it's an interesting spectacle to ask them to put the difference into words.

The technical definition lies in the substance's reaction to a shearing force:

Image

A solid will react by resisting the application of the force and statically deforming. A fluid, on the other hand, will not resist. Any shearing stress will therefore result in the motion of that fluid.

Breaking it down even further, the only difference between a liquid and a gas has to do with molecular cohesion, or, "how tightly packed the molecules are". A liquid, which consists of closely packed molecules with strong cohesive forces, will tend to remain constant in volume. A gas, on the other hand, consists of loosely-packed molecules with low cohesive forces, and as such, the gas is free to expand.





A Brief Overview of What We Won't Be Covering In Depth

There is a lot to say about the field of fluid mechanics without even having to jump into any math. First are foremost: How is it studied?

There are two ways to look at a problem in fluid mechanics:

1) You can stand back and watch the fluid flow as a whole, studying the distribution of pressure and velocity of the flow. This is called "eulerian" method.

2) You can plop yourself of the back of an individual molecule and take note of the changes in pressure (as opposed to the pressure as a whole). This is called the "lagrangian" method.

Typically, the eulerian method is used to in basic flow analysis.

A fluid problem has many properties that we take an interest in; that is, we want to see how these properties are affected as the fluid (or fluids) flows (or flow) from one end to another. For example, we may be interested in the pressure, density, temperature, or all three at once. There are other properties as well, such as internal energy, enthalpy, entropy, specific heats, viscosities, specific gravity, potential and kinetic energies, or thermal conductivity, but those are beyond the scope of this lecture.






A Not-So-Brief Overview of The Math

Math important. Math good. Math describe. Math show. Math challenging. Math rewarding. Math make us smart. Math make our ship go.



You think I'm joking, don't you.



There's a lot of math involved in fluid mechanics. A lot of complicated math. Fortunately, we won't be using most of it. I just want to introduce three simple mathematical relations:
    1) The relationship between pressure, density, and temperature of a gas
    2) The relationship between pressure and velocity of a fluid
    3) The relationship between area, pressure, and velocity in a pipe

1) You probably learned this one in high school chemistry. If you didn't take chemistry, or physics, or really an science whatsoever, then it might be foreign to you. It's a state relation for gasses, and it's called, for lack of a better term, the perfect gas law (NOTE: Some people refer to it as the ideal gas law, and the two are often used interchangeably. While there are some technical differences, and while the nomenclature may break down depending on how specific we want to get, we can, for now, assume that they pretty much mean the same thing).

The relationship is describes as such: "The pressure of a gas is proportional to the product of its density and temperature." That's big fancy-talk for this:

Image
where "P" = Pressure, that small "p" (called "rho") is the density, "T" is temperature, and "R" is a constant.

That's how a car engine pushes a piston, or why a car tire might explode on a hot summer day! (I'm of course simplifying these examples, but the basic principle still applies).

And that's the perfect gas law. It's interesting to note that there is no analogous relation for liquids. This is because liquids are largely considered to be "incompressible" (remember how earlier I stated that a liquids volume tends to remain unchanged?). This is why they are useful for things such as hydraulic lifts, etc.






2) The second mathematical relation also applies to gasses, but this time it's for gasses that are in motion! It's also the basic description for how, yes how (not just "why") airplanes fly.

Caveat: actually, there are two ways to describe how an airplane flies, but the end results are the same:

- The air flow over the top of a wing is faster, while the flow of air under the wing is slower. This results in lower pressure up top and higher pressure on bottom. This pressure differential creates a net upward force.

- The momentum of the incoming air is changed per Newton's Third Law of Motion; the incoming air exerts a force on the bottom of the wing, thus creating a net upward force. Due to the whole "for every action there is an equal and opposite reaction" matter, the wing simultaneously deflects the incoming air slightly downward.


But let's focus on the first description. This phenomenon is described by "Bernoulli's Principle", which states that there is a direct relationship between the velocity of a fluid and its pressure. More precisely, it describes a relationship between the energy of a fluid as a combination of pressure, kinetic, and potential energies, but this relationship only applies to steady, incompressible flow.

Image

But let's simplify that even further. Suppose there's no change in height. Now we can say simply this:

Image

That describes the relationship between pressure and velocity: For an increase in velocity, there is a corresponding exponential decrease in pressure. Here's a picture, because you like pictures:

Image

This is why you feel pulled towards the tracks when a fast train goes by; why the air over the top of the wing on an airplane moves faster.

And that's Bernoulli.


But what about Newton? Well, that's pretty easy to illustrate. IMAGINE: You're driving down the highway, and you open your window and stick your hand out flat into the wind. When you keep your hand level, nothing happens. But as soon as you start tilting it up, you feel the force of the incoming air hitting the bottom of your hand and pushing it up. That right there is a momentum exchange; that force you feel is the LIFT being generated by your hand. Also, the air pressure over the top of your hand is a lot lower than the pressure on bottom.

So what am I getting at here?

ImageImageImage

As you can see, there's a lot of contention for describing how an airplane flies. There's a lot of misinformation. Maybe I can help clear it up for you:

Both Newton and Bernoulli describe two parts of the same thing: LIFT is the product of a momentum exchange between the air and the wing which, by the nature of geometry and fluid flow, results in a pressure differential between the top and bottom of the wing.

In other words, the pressure differential is a result of the airflow interacting with the wing via Newton's Third Law. The curved shape of an airfoil simply helps to keep the airflow smooth and attached to the surface to promote better lifting characteristics and increase the maximum angle between the airflow and airfoil before it stalls (that is, before the airflow behind the wing becomes detached, super turbulent, and life-threatening). It's entirely possible to produce LIFT with a flat surface (many supersonic jet fighters have a flat plate for a wing. Just look at the Lockheed F-104 Starfighter), but since airplanes fly through such a wide array of conditions, they need to be designed to have the best lifting characteristics throughout all of its flight envelope.







3) The third mathematical relation I want to cover regards flow through a nozzle. Why is that important? Well, nozzles are everywhere. Garden hoses, shower heads, fuel injectors, airplane engines, and rocket engines are all nozzles. The relationship is simple: For a given fluid flow, a decrease in area corresponds to an increase in velocity. Here's a nozzle.

Image

Imagine you're standing outside (yeah right. Nerds like us standing outside? Give me a break!). Now imagine you're holding your garden hose and watering your plants. Now imagine putting your thumb over the end of the hose. What happens? You feel a strong resistance: the pressure inside builds up. Also water starts squirting everywhere, and now instead of drowning the plant in front of you, you can dowse your neighbor who's on the other side of the bushes.

So what happened? Your thumb turned into a nozzle! You decreased the cross-sectional area of the hose, and the water in turn reacted by going faster due to an increase in pressure.

"BUT WAIT!" you say. "Didn't you just say earlier that, according to Bernoulli's principle, an increase in velocity corresponds to a decrease in pressure??"

Why, yes I did. And you would be correct. The thing is, it still applies. Let's look at the situation in a before-and-after sense:

Before
-The water pressure far, far away, like, at the spicket, is high. The water pressure at the end is low. This is largely due to friction in the hose.
-The mass flow rate stays the same from beginning to end, but energy and momentum are lost due to friction in the walls.
- The hose diameter stays constant.
- Since the water pressure at the end of the hose is relatively small compared to the outside pressure, there is only a small drop in pressure and the velocity remains largely unaffected.


After
- You decrease the cross-sectional area of the hose, thereby increasing the water pressure
- You block the flow of most of the water, so the pressure is driven up. Now the pressure in the entire length of the hose is higher.
- The length of your thumb is far shorter than the length of the hose, so the momentum and energy lost as water travels past your thumb is relatively small.
- Since the water pressure at the end of the hose is now relatively LARGE compared to the outside pressure, there is a LARGE drop in pressure and the velocity will therefore increase dramatically.

Ta-daaaa!






But the reality is far, far more complicated. If you want to get accurate results, you have to be prepared to dig deep; that is, do some hard-ass math. Case in point: The Navier-Stokes Equations.

For the smooth flow of an incompressible fluid with constant density and viscosity (i.e. water ), you can describe the velocity in 3 dimensions using the Navier-Stokes Equations:

Image

Believe it or not, a form of these equations are used in video games to emulate liquid and gas dynamics. But of course, they are also used in Computational Fluid Dynamics to predict airflow patterns (i.e. air passing through city buildings; over an airplane, etc).


For every scenario, there is a different set of equations to model it and predict the outcome.
Flow in pipes? There's an equation for that.
Flow in pipes with heating? There's an equation for that.
Flow in a square pipe? There's an equation for that.
Flow in a triangular pipe? There's an equation for that.
Flow in an open channel? There's an equation for that.
Turbulent flow in pipes? There's an equation for that.
Flow over a flat plate? There's an equation for that.
Flow through a nozzle? There's an equation for that.
Supersonic flow? There's an equation for that.
Flow in a pipe with heat added, bends in the pipeline, rough walls, obstacles in the way, nozzles in the pipeline, and with multiple inlets and outlets? There's an equation for that.

But all of these are derived from the same basic set of principles that govern fluid mechanics:

1) Mass is conserved
2) Momentum is conserved
3) Energy is conserved






Since I mentioned in....Supersonic Flow!

I wanted to talk about this briefly since supersonic flow is a rather interesting (and relatively recent) field of study.

Remember when I talked briefly about nozzles? Well....now instead of water going through the hose, let's imagine it's air. So what happens if we have airflow inside a converging nozzle? Well, naturally, the velocity of the air will increase. And what if we have that same airflow inside a diverging nozzle? Well, naturally, the velocity of the air will decrease.

But how far can we take it? If we have a longer nozzle and a bigger change in area, will the air just continue to go faster and faster?

Turns out, for a nozzle that just converges, the airspeed will peak when it is going exactly sonic; that is, when the air flow is moving at Mach 1 compared to its static equivalent. This will also always occur at the end of the nozzle (called the throat: its narrowest part). But if the air is limited in speed for a nozzle, how can we get things like rocket engines to eject gasses at supersonic speeds?

The answer lies in a weird, almost contradicting phenomenon. For air that is moving at sonic (or faster) velocities, a diverging nozzle will actually INCREASE its velocity!! This increase in velocity is paid for at the expense of a corresponding decrease in pressure and temperature. So if we slap a diverging nozzle onto the end of a converging nozzle and drive up the pressure, the airflow will increase in velocity throughout the entire assembly. It will increase from 0 to Mach 1 in the converging section, going Mach 1 at the throat, and increase further from Mach 1 to Mach 2, 3, 4, 5, etc in the diverging section. WHAT THE ACTUAL F***!?

It's crazy, I know, but it's only by this principle can rocket engines produce enough thrust to lift a giant behemoth off the ground.

Confused? Here's a handy infographic:

Image

Don't even get me started about shockwaves, though.






So there you have it. A basic introduction to fluid flow, it's descriptive equations, some interesting phenomena associated with it, and how it affects your everyday life.

_________________
"We must question the story logic of having an all-knowing all-powerful God, who creates faulty Humans, and then blames them for his own mistakes."


Mon Mar 24, 2014 3:15 pm
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